The availability of a Wi-Fi Internet connection is not imperative for every application for the iPad. Some applications that you download to your tablet require a Wi-Fi connection or, in the case of some iPads, a 3G or 4G connection. Other applications function fully whether the iPad is connected to the Internet or not.
With tens of thousands of apps to browse and consider downloading from Apple's App Store, it's easy to find apps that do not require a connection to the Internet. If you download an address book application to help you manage your contacts, there's no immediate need to have an Internet connection. For this style of app, you'd only need a connection if you wished to contact someone by clicking on his email address in the app.
Many kinds of iPad apps require a connection to the Internet. These apps include topics such as weather, news and sports. For example, an app that gives you updated weather forecast information requires a connection so that it can retrieve the latest data. Without a connection, the app only shows the data it loaded the last time it could reach the Internet. Likewise, a sports app that displays updated scores requires an Internet connection.
Some apps don't need the Internet to function, but having the Internet available augments them. This can commonly be said for iPad games. Angry Birds, for example, is a commonly downloaded game from the App Store. You don't need the Internet to play it, but you require a connection to download perks, upload your score to the game's hall of fame and compare your score with your friends' scores.
If it's important for you to know how dependent an app is on a network connection, you can do so by reading the description of the app either by browsing through the App Store icon on your iPad or doing the same on a computer. When you click on an app, scroll down to read the details of it. In the case of games, for example, the inclusion of the term "multiplayer" means the app needs the Internet.